Neharis theorem for convex domain Hankel and Toeplitz operators in several variables

We prove Neharis theorem for integral Hankel and Toeplitz operators on simple convex polytopes in several variables. A special case of the theorem, generalizing the boundedness criterion of the Hankel and Toeplitz operators on the Paley-Wiener space, reads as follows. Let $Xi = (0,1)^d$ be a $d$-dimensional cube, and for a distribution $f$ on $2Xi$, consider the Hankel operator $$Gamma_f (g)(x)=int_{Xi} f(x+y) g(y) , dy, quad x inXi.$$ Then $Gamma_f$ extends to a bounded operator on $L^2(Xi)$ if and only if there is a bounded function $b$ on $mathbb{R}^d$ whose Fourier transform coincides with $f$ on $2Xi$. This special case has an immediate application in matrix extension theory: every finite multi-level block Toeplitz matrix can be boundedly extended to an infinite multi-level block Toeplitz matrix. In particular, block Toeplitz operators with blocks which are themselves Toeplitz, can be extended to bounded infinite block Toeplitz operators with Toeplitz blocks.